Is the following true?
For real-valued, full-rank matrices $A_{nxp}$, $B_{pxn}$ with $rank(A)=rank(B)=p$, we know that $r(AB)=p$.
If true, please let me know how to prove it (and what properties are needed), and if not, could you please help by demonstrating a simple counterexample?
Edit: $n>p$, as the application here is linear models.
Yes it is true. You can view A, B as linear transformations, so $B$ would be a surjective map from $\mathbb R^n \rightarrow \mathbb R^p$, meanwhile $A$ would be an injective from $\mathbb R^p \rightarrow \mathbb R^n$. Their composition is a map from $\mathbb R^n \rightarrow \mathbb R^n$, while the range would be the same as $\text{range}(A) $, which is a $p$-dimensional space.